Convergence of Adaptive Finite Element Methods

Abstract

We develop adaptive finite element methods (AFEMs) for elliptic problems, and prove their convergence, based on ideas introduced by D\"{o}rfler \cite{Dw96}, and Morin, Nochetto, and Siebert \cite{MNS00, MNS02}. We first study an AFEM for general second order linear elliptic PDEs, thereby extending the results of Morin et al \cite{MNS00,MNS02} that are valid for the Laplace operator. The proof of convergence relies on quasi-orthogonality, which accounts for the bilinear form not being a scalar product, together with novel error and oscillation reduction estimates, which now do not decouple. We show that AFEM is a contraction for the sum of energy error plus oscillation. Numerical experiments, including oscillatory coefficients and {both coercive and non-coercive} convection-diffusion PDEs, illustrate the theory and yield optimal meshes. The role of oscillation control is now more crucial than in \cite{MNS00,MNS02} and is discussed and documented in the experiments.

We next introduce an AFEM for the Laplace-Beltrami operator on $C^1$ graphs in $R^d ~(d\ge2)$. We first derive a posteriori error estimates that account for both the energy error in $H^1$ and the geometric error in $W^1_\infty$ due to approximation of the surface by a polyhedral one. We devise a marking strategy to reduce the energy and geometric errors as well as the geometric oscillation. We prove that AFEM is a contraction on a suitably scaled sum of these three quantities as soon as the geometric oscillation has been reduced beyond a threshold. The resulting AFEM converges without knowing such threshold or any constants, and starting from any coarse initial triangulation. Several numerical experiments illustrate the theory.

Finally, we introduce and analyze an AFEM for the Laplace-Beltrami operator on parametric surfaces, thereby extending the results for graphs. Note that, due to the nature of parametric surfaces, the geometric oscillation is now measured in terms of the differences of tangential gradients rather than differences of normals as for graphs. Numerical experiments with closed surfaces are provided to illustrate the theory.